Computational Methods for Debonding in Composites

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272 M. Kästner et al. In order to locate a given interface in the domain of the X-element the signed distance ϕi between the interface and node i is computed whereat the sign of ϕi depends on the position of the node with respect to the interface [16]. The distribution ϕ(x) in the element can be interpolated using the standard shape functions (13.39) ϕ (x)=∑ i The position of a material interface Γ is then given by ϕ(x)=0. Due to the interpolation which limits the accuracy of the interface approximation the mesh has to be sufficiently refined in order to locate an interface precisely. The interpolated distance function ϕ(x) is of particular importance for the formulation of enrichment functions. At a material interface a sudden change of coefficients of the partial differential equations occurs. This implicates a discontinuity of strain fields perpendicular to the interface as described in equation (Eq. (13.37)). Looking at the interpolated values of the distance function ϕ(x) it can be seen that the function itself and its first partial derivatives are continuous over the interface. Using the absolute value of the interpolation (Eq. (13.39)) is a smart way to introduce the desired discontinuity [16] � � � � ˜F � � (ϕ (x)) = � Niϕi� (13.40) � On the other hand the influence of the enrichment function ˜F is not limited to the region of a single element which is unfavorable for the definition of a special X-element that is to be used together with standard elements in the same mesh. In this case an enrichment function that is zero at element boundaries connected to standard elements is desired. As suggested in [12] this can be ensured by choosing the enrichment function � � � F (ϕ (x)) = ∑Ni |ϕi|−� i �∑ � � � Niϕi� (13.41) i � which is plotted for a straight interface intersecting a 2D plane element in Fig. 13.8. After all essential data such as material properties, the location of the interface in each element defined by the distance function values ϕi at nodes, etc. are obtained during pre-processing, the integration of the element stiffness matrix K e � = B T DBdΩ e (13.42) Ω e � ∑ i Niϕi is performed where D is a matrix representation of Cijkl. Due to the enrichment the matrix B contains partial derivatives of the enrichment function which are discontinuous over the interface as well as different material properties. In order to handle the discontinuous integrand, the element domain Ω e is
13 Effective Stiffness Properties of Textile-Reinforced Composites 273 Fig. 13.8 Plot of the enrichment function F (ϕ (x)) over a 2D plane element domain Fig. 13.9 Integration subdomains F(x,y) 1 0.5 0 0 0.25 0.5 0.75 x 1 Material 1 Material 2 1.25 0 Integration subdomain divided into ns integration subdomains Ω s i applying DELAUNAY hex-tet subdivision in three dimensions � dΩ e ns � ⇒ (13.43) as shown in Fig. 13.9. After the decomposition each subdomain is assigned with the right material properties and the integration of the element stiffness matrix is carried out using standard GAUSS quadrature rules. Finally, the element stiffness matrix is passed into the commerical FE-code where the global stiffness matrix is assembled. When modelling textile-reinforced composites – mainly due to the combination of high fibre volume fractions and complex reinforcing architectures – the problem of branching material interfaces is observed. As shown in Fig. 13.10a the material interface between warp and weft yarn opens into two yarn-matrix interfaces. Therefore, a second new element type called 2X-element is introduced which can handle two branching material interfaces in a single element domain. Numerically the case of two material interfaces in a single element is treated by introducing another set of additional degrees of freedom bi at nodes whose support is cut by both interfaces leading to a further enrichment of the displacement approximation u X-FEM = ∑ i Niui +∑ j ∑ i=1 Nja jF1 (ϕ1(x)) +∑ k dΩ s i 0.5 y 1 1.5 NkbkF2 (ϕ2(x)). (13.44)
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Mechanical Response of Composites
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Pedro P. Camanho • C.G. Dávila S
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Preface The methodology for designi
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Acknowledgements The Editors of thi
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x Contents 3 Practical Challenges i
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xii Contents 8.2.4 Modelling Effect
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xiv Contents 13.1.2 EffectiveProper
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xvi List of Contributors Clara Schu
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Chapter 1 Computational Methods for
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1 Computational Methods for Debondi
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28 R. Rolfes et al. functions by me
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30 R. Rolfes et al. Microscale fibe
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32 R. Rolfes et al. symmetric, whic
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34 R. Rolfes et al. Fig. 2.5 Discre
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36 R. Rolfes et al. Unit cell (a) S
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38 R. Rolfes et al. stress state. T
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40 R. Rolfes et al. biaxial compres
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42 R. Rolfes et al. Damage Evolutio
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44 R. Rolfes et al. Fig. 2.14 Radia
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46 R. Rolfes et al. Table 2.2 Elast
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48 R. Rolfes et al. A dev is the de
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50 R. Rolfes et al. uniaxial compre
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52 R. Rolfes et al. Stress in MPa -
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54 R. Rolfes et al. 2.4.2 Results o
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56 R. Rolfes et al. 12. Hughes TJR
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58 B.N. Cox et al. inform models; a
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60 B.N. Cox et al. 3.2 The Structur
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62 B.N. Cox et al. be successfully
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64 B.N. Cox et al. high fluxes of c
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66 B.N. Cox et al. provides poor in
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68 B.N. Cox et al. For example, a c
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70 B.N. Cox et al. failures in comp
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72 B.N. Cox et al. mixed-mode crack
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74 B.N. Cox et al. 24. Elices M, Gu
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Chapter 4 Analytical and Numerical
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4 Analytical and Numerical Investig
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100 C. Schuecker and H.E. Petterman
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Chapter 6 Study of Delamination in
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6 Study of Delamination with in Com
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Chapter 7 Interaction Between Intra
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7 Interaction Between Intraply and
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Chapter 8 A Numerical Material Mode
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8 A Numerical Material Model for Pr
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180 N. Zobeiry et al. There are sev
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182 N. Zobeiry et al. for construct
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184 N. Zobeiry et al. In compressio
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186 N. Zobeiry et al. c = 38.7 mm h
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188 N. Zobeiry et al. displacement
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190 N. Zobeiry et al. No Notched St
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192 N. Zobeiry et al. where g f is
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194 N. Zobeiry et al. References 1.
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Chapter 10 Elastoplastic Modeling o
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10 Elastoplastic Modeling of Multi-
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